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Theorem inawina 8967
Description: Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
inawina  |-  ( A  e.  Inacc  ->  A  e.  InaccW )

Proof of Theorem inawina
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfon 8534 . . . . 5  |-  ( cf `  A )  e.  On
2 eleq1 2526 . . . . 5  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . . . 4  |-  ( ( cf `  A )  =  A  ->  A  e.  On )
433ad2ant2 1010 . . 3  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  On )
5 idd 24 . . . 4  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  A  =/=  (/) ) )
6 idd 24 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  =  A  -> 
( cf `  A
)  =  A ) )
7 inawinalem 8966 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
85, 6, 73anim123d 1297 . . 3  |-  ( A  e.  On  ->  (
( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) ) )
94, 8mpcom 36 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
10 elina 8964 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
11 elwina 8963 . 2  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
129, 10, 113imtr4i 266 1  |-  ( A  e.  Inacc  ->  A  e.  InaccW )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799   (/)c0 3744   ~Pcpw 3967   class class class wbr 4399   Oncon0 4826   ` cfv 5525    ~< csdm 7418   cfccf 8217   InaccWcwina 8959   Inacccina 8960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-recs 6941  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-card 8219  df-cf 8221  df-wina 8961  df-ina 8962
This theorem is referenced by:  gchina  8976  inar1  9052  inatsk  9055  tskuni  9060  grur1a  9096  grur1  9097  inaprc  9113
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